The sequence $(x_n)$ is given by $x_1=1$ and $$x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1.$$(a) Prove that $x_{n+1}\ge x_n$ for all $n\ge4$. (b) Prove that $\lfloor x_n\rfloor=n$ for all $n\ge4$.
Source: Ukraine 1998 Grade 11 P8
Tags: floor function, Sequences, algebra
The sequence $(x_n)$ is given by $x_1=1$ and $$x_{n+1}=\frac{n^2}{x_n}+\frac{x_n}{n^2}+2\enspace\text{for }n\ge1.$$(a) Prove that $x_{n+1}\ge x_n$ for all $n\ge4$. (b) Prove that $\lfloor x_n\rfloor=n$ for all $n\ge4$.